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However, there is a second mechanism by which physical systems generate scale-free correlations, and this is by tuning parameters to a critical point ( 8, 13). Now there are no Goldstone modes, because choosing a speed does not correspond to breaking any plausible symmetry of the system. Not just the fluctuations in flight direction, but also the fluctuations in flight speed are correlated over long distances ( 6). We have shown that this is more than a metaphor ( 12): the minimally structured model consistent with the observed correlations among flight directions of neighboring birds is equivalent to a model of spins in a magnet, and the resulting (parameter-free) prediction of long-ranged correlations among fluctuations in flight direction agrees quantitatively with the data. If we can think of the alignment of flight directions in a flock as being like the alignment of spins in a magnet ( 9– 11), then we can understand the emergence of scale-free correlations via Goldstone’s theorem. If the system spontaneously breaks a continuous symmetry, for example when all of the spins in a magnet select a particular direction in space along which the macroscopic magnetization will point, then the fluctuations in the system are dominated by Goldstone modes that do not decay on any fixed length scale ( 8). In physics, we have two very different mechanisms for local interactions to produce long–ranged correlations. If this is correct, then we have to understand how local interactions can generate correlations over much longer distances. It is generally believed that the interactions among birds in a flock are local-each bird aligns its flight direction and speed to those of its near neighbors ( 7). Strikingly, observations on flocks of starlings show that these correlations extend over very long distances, comparable to the size of the flock itself ( 6). Even in the absence of predators, we can see deviations of individual behavior from the average behavior of the flock, and correlations in these fluctuations provide a signature of information flow through the flock. Although it is difficult to measure this information flow directly ( 1), we know that attacks by predators on a flock have very low success rates ( 2– 4), and that the evasion of predators by starling flocks is associated with the triggering and propagation of waves through the flock ( 5). The entire flock must respond to dangers that may be visible only to a small fraction of individuals, requiring information to propagate over long distances. However, this average behavior is not enough for flocking to be advantageous. In a flock of birds, thousands of individuals will fly in the same direction and at the same speed, for long periods of time. In biological terms, criticality allows the flock to achieve maximal correlation across long distances with limited speed fluctuations. These models are mathematically equivalent to statistical physics models for ordering in magnets, and the correct prediction of scale-free correlations arises because the parameters-completely determined by the data-are in the critical regime. These minimally structured or maximum entropy models provide quantitative, parameter-free predictions for the spread of correlations throughout the flock, and these are in excellent agreement with the data.
Here we construct models for the joint distribution of velocities in the flock that reproduce the observed local correlations between individuals and their neighbors, as well as the variance of flight speeds across individuals, but otherwise have as little structure as possible.
Quantitative measurements on flocks of starlings, in particular, show that these fluctuations are scale-free, with effective correlation lengths proportional to the linear size of the flock. It is not just that thousands of individuals fly, on average, in the same direction and at the same speed, but that even the fluctuations around the mean velocity are correlated over long distances. Flocks of birds exhibit a remarkable degree of coordination and collective response.